Quantum steam tables. Free energy calculations for H2O

Quantum steam tables. Free energy calculations for H20, 0 20, H2S, and H2Se by adaptively optimized Monte Carlo Fourier path integrals

Robert O. Topper,a) Oi Zhang, Vi-Ping Liu, and Donald G. Truhlar Department of Chemistry and Supercomputer Institute, University of Minnesota, Minneapolis, Minnesota 55455-0431

(Received 12 October 92; accepted 3 December 92)

Converged quantum mechanical vibrational-rotational partition functions and free energies are calculated using realistic potential energy surfaces for several chalcogen dihydrides (H20, D20, H 2S, H2Se) over a wide range of temperatures (600-4000 K). We employ an adaptively optimized Monte Carlo integration scheme for computing vibrational-rotational partition functions by the Fourier path-integral method. The partition functions and free energies calculated in this way are compared to approximate calculations that assume the separation of vibrational motions from rotational motions. In the approximate calculations, rotations are treated as those of a classical rigid rotator, and vibrations are treated by perturbation theory methods or by the harmonic oscillator model. We find that the perturbation theory treatments yield molecular partition functions which agree closely overall (within ~ 7%) with the fully coupled accurate calculations, and these treatments reduce the errors by about a factor of 2 compared to the independent-mode harmonic oscillator model (with errors of ~ 16% ). These calculations indicate that vibrational anharmonicity and mode-mode coupling effects are significant, but that they may be treated with useful accuracy by perturbation theory for these molecules. The quantal free energies for gaseous water agree well with previously available approximate values for this well studied molecule, and similarly accurate values are also presented for the less well studied D20, H 2S, and H 2Se.

I. INTRODUCTION

In order to calculate equilibrium constants by molecular statistical mechanics and absolute reaction rates by transition state theory, it is necessary to calculate the canonical vibrational-rotational partition function Q( T) as a function of the temperature T from the potential energy function for isolated reactants, products, and transition states in the gas phase. This is directly equivalent to estimating the absolute free energy G; in particular, the relationship between the molecular partition function and the free energy of an ideal gas of N x molecules is given in the Born-Oppenheimer approximation byl-4

G= -NxkBT In Q(T)Qtrans(T) .N x

(1)

where kB is Boltzmann's constant, and Qtrans(T) is the translational partition function

3 (NxkBT) Qtrans(T)=it p. , (2)

it=t7T~:BT , (3)

with M the molecular mass and P the pressure. Recently, we presented a new Monte Carlo methodS

based on the Fourier path-integral formalism6-11 for the computation of molecular vibrational-rotational partition

·)Present address: Department of Chemistry, University of Rhode. Island, Kingston, RI 02881-0801.

functions, and we demonstrated its convergence properties by applying it to a model coupled oscillator problem5 and to the diatomic molecule HCI described in three Cartesian degrees of freedom. 12 In the present work, we apply the new Fourier path-integral Monte Carlo method to the computation of partition functions for several triatomic molecules: H20, D20, H2S, and H2Se. No approximate mode decouplings are invoked and the calculations represent the result of converged quantum mechanics for the assumed potential functions. We compare the results of these calculations to approximate calculations based on assuming separability of the vibrational and rotational degrees of freedom combined with treating the vibrations with harmonic and anharmonic models and the rotations with the standard classical rigid rotator model.

Although previous studies have made systematic comparisons between accurate quantum mechanics and approximate forms for partition functions for diatomic molecules at high temperatures,12-16 as well as for free energies for van der Waals clusters at low temperatures,lO,11 fewer studies have been carried out for polyatomic molecules,17-19 primarily because of the lack of accurate calculations of polyatomic energy levels. 18,20 In particular, we note that the molecular internal partition function is conventionally obtained by calculating the vibrationalrotational eigenenergies En using variational methods2,18,21 and summing Boltzmann factors to obtain the partition function byl-4

Q(T) = L exp( -f3En) , (4) n

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4992 Topper et al.: Quantum steam tables

where n designates all of the quantum numbers for each vibration-rotation state and /3= lIkBT. If the system of interest is studied at low temperature, an accurate calculation of the molecular zero-point energy for a small range of rotational excitation energies, and perhaps the excitation energies of a few low-frequency modes, may suffice to converge Eq. (4). However, if one needs to carry out calculations at higher temperatures (in the combustion regime, e.g.), if there are low-frequency internal rotations in the molecule, or if more than one conformer is thermally accessible, it may be very difficult to converge Eq. (4).

In those cases where Eq. (4) is difficult to converge, one may employ one of several approximation schemes which have been developed for partition functions. I-5,12-17,22-27 One goal of the present paper is to use the Fourier path-integral Monte Carlo method to test the accuracy which may be achieved by such schemes. Notably, almost all of the approximate methods assume that the vibrational motion is separable from the rotational motion, but this assumption is not made in the Fourier pathintegral scheme. Another goal of the present paper is to demonstrate the convergence properties of our Fourier path-integral Monte Carlo algorithm for triatomic molecules. Finally, we will present computed Gibbs free energy functions for four triatomic molecules. Note that the Gibbs free energy for H2Se is not tabulated in the JANAF tables28 of thermodynamic functions, and other previous estimates are highly approximate.

Section II briefly reviews the Fourier path-integral formalism and the adaptively optimized Monte Carlo method. Section III presents the coordinate system and details of the parameters used in the Fourier path-integral calculations. Section IV presents the potential surfaces used in the present study. Section V presents the approximate methods to be compared to the accurate calculations. Section VI presents the results of the calculations. Section VII discusses these results and Sec. Vln contains a summary bf our conclusions.

II. THE FOURIER PATH-INTEGRAL MONTE CARLO METHOD

Partition functions for molecular systems are calculated in the present paper by the Fourier path-integral Monte Carlo method. We use the Feynman path-integral treatment of quantum statistical mechanics,7 and the nuclei are treated as particles which move under the influence of a Born-Oppenheimer potential energy surface. Under this set of assumptions, the vibrational-rotational partition function for a polyatomic molecule may be written as the following path integral:

Q( T) =~ f dx f: YJ [x(s)]

xexp! -~ f:n ds H[X(S),P(S)]}, (5)

where x is a set of Cartesian coordinates which locate the nuclei in a center-of-mass frame, a is the usual rotational symmetry number,I,3,4 and f~§[x(s)] indicates an integral over all paths, which are denoted xes) (where s is a parameter with the units of time) that begin and end at points x in the N-dimensional molecular configuration space (Le., all closed paths), and H[x(s),p(s)] is the Hamiltonian for the system. The Hamiltonian is also a function of the associated path momenta given by pes). There is a formal correspondence between the parameter s in Eq. (5) and an imaginary time t that may be associated with each path in the path-integral representation of quantum dynamics.6

As discussed by Feynman,6,7 Miller,8 and Freeman, Doll, and co-workers,9-11 the paths may be expanded in Fourier series about free particle "reference paths" with no loss of generality. The exact partition function is then transformed from a path integral over all closed orbits to an infinite-dimensional Riemann integral; for a system with N internal degrees of freedom, this yields

Q(T)=J(T)~feo_ ···feo (fi dXj) ( fi IT daj,l)exp[- £ i: 2~-S(X,a)]. a -eo -eo 1=1 )=1 1=1 1=1 1=1 1,1

(6)

Here J( T) is the Jacobian of the transformation and it. is given by

(7)

a j,l are the coefficients used to represent the closed paths in a Fourier expansion, Le.,

(8)

The function S(x,a) is an action integral corresponding to

a path that starts at a point x in configuration space and ends at the same point after the complex time interval /3-1Z,

1 r/3n S(x,a) =1i J 0 ds V[x(s)]. (9)

The parameters aj,l are functions of the temperature and of the reduced masses f.L j associated with the N Cartesian variables x j in the center-of-mass coordinate system. These parameters are given by

(10)

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Topper et al.: Quantum steam tables 4993

In order to evaluate Eq. (6), which is an infinitedimensional integral over an infinite domain, we represent it in a form which can be evaluated by Monte Carl029

methods. This is accomplished5 by dividing the partition function for the fully coupled system by the partition function for a particle in an N-dimensional hypersphere with zero potential energy. For numerical work, the domain in configuration space in Eq. (6) is truncated to the interior of the same hypersphere, which is designated by D, and the infinite number of Fourier coefficients is truncated to a finite number (K) of coefficients per degree of freedom. The molecular partition function is then given by an integral over an [N + (NXK)]-dimensional domain:

Q( T) ~ Qph( T) J dx Joo ... Joo da (7 D -00 -00

Xexp[-S(x,a)]g(a), (11)

where Qph(T) is the partition function for a particle in an N-dimensional hypersphere, i.e.,

(12)

'rkN) is the hypervolume of the N-dimensional hypersphere, calculated from the standard formula3o

,,!{/2RN 'r(N)

h =r[(N/2)+I] , (13)

where r(N) is the usual gamma function, and R is the hyperradius of the hypersphere. The function g(a) is a normalized (NXK)-dimensional Gaussian probability density function in Fourier coefficient space given by

G(a) g(a) =--::-----:;-----

J~oo'" J~oo da G(a) (14)

with

G(a) =exp { - il /#1 2~J (15)

If K and D are sufficiently large, Eq. (11) represents the accurate quantum mechanical result to arbitrary accuracy. We evaluate Eq. (11) as described previously.5,12 The nuclear coordinates are sampled using adaptively optimized stratified sampling (AOSS) of strata defined by concentric hyperspheres within D, and the Fourier coefficient degrees offreedom are sampled according to g(a) by using the Box-Muller algorithm31,32 to generate an uncorre1ated sample in the Fourier coefficient space. This combination of techniques has been referred to as the AOSS-U method, where U indicates that the samples are sequentially uncorrelated.s

The configuration space coordinates of each sample are confined within the stratum boundaries by using rejection29

methods. For example, to generate samples within a hypersphere, samples are generated within an N-dimensional hypercube just enclosing the hypersphere, and all samples which fall outside of the hypersphere are ignored, i.e., not evaluated or accumulated. As has been discussed previ-

ously,5 this procedure rapidly becomes inefficient as N increases because the ratio of the volume of the sphere to that of the cube enclosing it tends rapidly towards zero as a function of increasing N. However, we have found that for small molecules, the computational cost of generating the unused configuration space points is very small compared to the cost of integrating the potential along the sampled paths, and therefore the present procedure is useful in a practical sense for these systems. The Monte Carlo estimate of the integral is then given by5,!2

3 [Qph(T) h nk

(Q(T)n= I I exp[-S(xik,aik)], k=! nk ik=!

(16)

where nk is the number of samples within each stratum, n

is the total number of samples, and [Qph ( T) h is given by Eq. (12), but with the volume of the kth stratum ('rk )

substituted for 'r. We remind the reader that, as discussed previously,5

one of the advantages of the present sampling scheme is that it is fully uncorrelated, and so the statistical error may be estimated reliably and in a straightforward manner. The statistical error of the partition function of Eq. (16) is denoted Wn and is given by

~ [Qph(T)]~ [( -2S(x,a» _( -S(x,a»2] W n= £.., nk- 1 e nk e nk '

k=! (17)

In the present work, the coordinate system used to describe the configuration space is different than the ones used in previous work. In the following sections, we discuss

.. the coordinate system and potential surfaces used in the present study in some detail.

III. COORDINATE TRANSFORMATIONS AND IMPLEMENTATION OF THE FOURIER PATH-INTEGRAL METHOD

We will consider the vibrational-rotational dynamics of several triatomic molecules. In order to apply the Fourier path-integral method summarized above, we specify the nuclear positions of the triatomic in a coordinate system defined such that the kinetic energy operator is written in the form

N ~2

T=~ I Pj, 2 j= 1 f-£ j

(19)

where N is the number of vibrational-rotational degrees of freedom (i.e., N=6), Pj is the momentum operator for degree of freedom j, and f-£ j is a reduced mass. The origin of the coordinate system is fixed at the molecular center of mass.

These requiremepts are satisfied by the usual Jacobi coordinates.33-38 For a triatomic molecule (see Fig. 1) with nuclear masses (rnA' mE' me), the transformation between Cartesian coordinates z and Jacobi coordinates S is

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z

x A =O,S,Se

FIG. 1. The coordinate system used in the present study. The relationship between the Cartesian coordinates z and Jacobi coordinates S is discussed in the text. The inset shows the internal coordinates used to evaluate the potential energy along each sampled path.

SJ+6= (mAzJ+m BZJ+3 +mczJ+6)/M,

with J= 1,2,3. Here (Zl,Z2,Z3) are the Cartesian coordinates of A with respect to a laboratory frame, (z4"",z9) are similarly defined for atoms Band e, respectively, and M is the total mass. (S1,S2,S3) specify the location of B relative to e, (S4,S5,S6) locate A relative to the Be center of mass, and (S7,S8,S9) locate the molecular center of mass. In this coordinate system, f-L j are given by

and

f-L4 = f-L5 = f-L6

mBmC

(mB+mC) ,

f-L7=f-LS=f-L9=M.

(21)

(22)

(23)

In order to simplify the Hamiltonian and treat all coordinates on an equal dynamical footing, we apply a transformation to mass-scaled Jacobi coordinates {x}, i.e.,

Xj= (f-L/f-L) 1I2Sj, (24)

where f-L is the scaling mass. All physical results are independent of f-L, so we set it equal to 2500 a. u. of mass. Fixing the origin at the molecular center of mass, the barycentric Hamiltonian becomes

6

H=(2f-L)-1 L p;,+V(X[,,,,,X6)' j=1 J

(25)

Using Eqs. (20)-(24), one can represent the canonical transformation between Cartesian and mass-scaled Jacobi coordinates as a 9 X 9 matrix T, such that

z=Tx. (26)

In the Fourier path-integral calculations, we set X7' x8, and X9 equal to zero and work in the remaining sixdimensional space specified by x1,,,,,x6' We draw the configuration space starting point for each sampled path uniformly from the interior of a six-dimensional hypersphere. The Fourier coefficients for the sampled path are drawn from a multidimensional Gaussian distribution via the Box-Muller algorithm.31,32 The matrix T is then used to transform each configuration along each sampled path to Cartesian coordinates, which are then used to calculate the potential energy along the sampled paths.

The masses are taken as mH=1837.15me, mD = 3671.49me, mo=29157.0me> ms=58281.6me, and mSe

= 145679me, where me is the atomic unit of mass (i.e., the mass of an electron). The parameters used in the Fourier path-integral calculations are specified below in Sec. VI.

IV. MOLECULAR POTENTIAL ENERGY SURFACES

In this study, we use a series of potential surfaces develqpeslby Kauppi and Halonen39 for covalently bound triatomic molecules, and also one surface presented by Zhao et al. 40 A single functional form is used with a different set of parameters for each case. The internal coordinates of the molecule are specified by (see Fig. 1)

R1 = ~(Z6-Z3)2+ (Z5-Z2)2+ (Z4-Z1)2, (27)

(28)

cP=cos-1 [~:R~2]. (29)

Then a set of auxiliary variables (Y1'Y2,O) is defined by

Yi= 1-exp [ -a(Rj-Re)],

O=CP-CPe,

(30)

(31)

with Re the value of R j at the equilibrium configuration and CPe the equilibrium internal bond angle. The potential energy expressed as a function of these internal coordinates was given by Halonen and Carrington41 in the form

(32)

(33)

(34)




(35)

For the first four cases, the functional form given in Eqs. (27)-(35) is used with the parameters originally given by Kauppi and Halonen for H20, D 20, H 2S, and H 2Se, and in the fifth and sixth cases, we studied H20 and D 20 with a modified set of parameters40 developed for the study of microhydrated reactions.40,42,43 The modified parameters still provide good agreement with the experimental harmonic frequencies of the water molecule, but they also can be used to represent accurately the properties of the CI-(H20) and Cl-(D20) ionic complexes. For convenience, the parameters used for all cases are summarized in the supplementary material.44

V. APPROXIMATE METHODS FOR PARTITION FUNCTION CALCULATIONS

In the present work, we compare partition functions calculated by Fourier path-integral Monte Carlo methods to those calculated by two ways of using perturbation theory and by the harmonic approximation. The usual starting place for approximate calculations is to assume separation of the vibrational motion from the rotational motion, and in this approximation, the partition function is a product of vibrational and rotational factors t-4

(36)

The equations in the rest of this section assume that the molecule is nonlinear and there are no degenerate vibrations, both of which are true for the cases studied in this paper.

Under the assumption that the rotations are those of a classical rigid body with internuclear distances constrained to those of the equilibrium geometry, the partition function for a rigid polyatomic molecule with principal moments of inertia 1 A' 1 B' and 1 C and symmetry number (T is given by t-4,45

1T1/2 (21 ) 1/2 Q~?-(T) =~ IT {3n~ ,

a=A,B,C

(37)

where "CRR" designates the classical rigid rotator approximation. The CRR approximation to the rotational degrees of freedom was recently tested20 against a nonrigid quantum-mechanical treatment of the rotational degrees of freedom for the water molecule. It was found to underestimate the partition function by 2% at 500-1000 K and 3.5% at 2000 K.

The approximate methods discussed in the present paper all assume Eqs. (36) and (37), but they use different expressions for the vibrational partition function. The most common approximation to the vibrational partition func-

tion is to treat the vibrations as independent-normal-mode harmonic oscillations. The partition function for this model is given in closed form as3,4

o N-3 exp( -{3fzOJm/2)

~b (T)= mIlt 1-exp(-{3fzOJm ) ' (38)

where OJm is the vibrational frequency (in radians per second) associated with normal mode m.

Another way to approximate the vibrational partition function is to carry out approximate calculations of the energy levels using perturbation theory23,46-54 and sum their Boltzmann factors according to Eq. (4), replacing n by the set of quantum numbers {Vi} for each vibrational energy level. In the present paper, we carry out a fully coupled perturbation treatment of the J=O vibrational energy levels, where J is the rotational quantum number. We use standard methods4

6-52 to calculate the perturbation theory vibrational energy levels using a quartic approximation to the various force fields. As usual, our perturbation calculations include all terms up to second order in the cubic force constants and up to first order in the quartic force constants. We used the SUR VIBTM computer program, developed by Ermler, Hsieh, and Harding,53 with the potential energy surfaces presented in Sec. IV as input for the computations. No corrections for Fermi or other resonances were included.

We next describe briefly the perturbation theory calculations. We define a set of reduced (dimensionless) normal coordinates q m'

(21TCJ.1/iim) 1/2 .

qm= n Um, (39)

where U m is a mass-scaled normal-mode coordinate related to the usual55 mass-weighted normal coordinate Qm by

(40)

and vm=OJin/21Te is the spectroscopic frequency in units of cm- t • Note that um in Eqs. (39) and (40) is a linear combination of x j of Eq. (24) with unitless coefficients. Carrying out a fourth order Taylor expansion of the potential energy function, we can write an approximate form for the potential of a nonlinear molecule as a function of the reduced normal mode coordinates

he N-3 _ 2 he -V;:::; Ve+ 2! m~t vmqm+ 3! m~o f mnoqmqnqo

he +4!

m,n,o,p (41 )

wher~ Ve is the energy at the eqUilibrium geometry, f mno

and f mnop are third and fourth order normal mode force constants, respectively, and the summations are unrestricted, i.e., all sums are from 1 to N - 3. We note that Eq. (41) uses only one of many possible conventions for the Taylor expansion, and that care must be used when comparing to other discussions of perturbation theory for vibrational energy levels in the literature. Throughout the present paper, we place the zero of energy at the bottom of each potential well so that Ve=O.

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4996 Topper et at.: Quantum steam tables

Through the use of standard second order perturbation theory, the vibrational energy levels, i.e., the energy levels for a molecule with rotational quantum number J=O, are then given by

N-3 1) E[{vmll=Ve+Eo+hc m~1 Vm(Vm+ 2

+hc m~n Xmn( Vm+~) (Vn+~)' (42)

where xmn is a second order anharmonicity constant, and Eo is a constant (i.e., independent of vibrational quantum numbers) term which arises in the perturbation theory treatment. Note that Eo is often neglected in spectroscopic applications, where only energy level differences are computed; however, it is necessary in the calculation of perturbation theory energy levels for thermochemical applications, as its omission noticeably affects the absolute values of the vibrational energy levels, and therefore the vibrational partition functions.23 This term depends on the force constants, rotational constants, and Coriolis constants23,50, 5 I

-:2 1 ~ - 7" fmmm 3 ~

EoIhc=64 '" f mmmm-576 '" ---+ 64 '" m m Vm m¥=n

1 L 4 m<n<o

VmVnVoJ~no Dmno

where

Dmno= (vm+vn+vo) (vm-vn-vo) (-vm+vn-=-vo) ~.

x (-vm-vn+vo), - (44)

a designates a principal rotational axis (A, B, or C), (;;::; is a Coriolis constant56 in units of cm -I, and B~a) is a rotational constant for the equilibrium configuration (also in units of cm- I

).

Given the approximate expression for the potential function in Eq. (35), the anharmonicity constants for an asymmetric top molecule are given as47,49,50

- ~ -2 V m - Vn 1 1 N-3 [8-2 3-2 ]

Xmm=16fmmmm-16 .-; fmmn vn(4v~-~) (45)

_~ f- _" f mmo fnno Xmn - 4 mmnn '" 4v

o 0

(46)

In full vibrational perturbation theory calculations, the vibrational partition function is evaluated by summing the Boltzmann factors of the energy levels given in Eq. (42)

(47) m

where m denotes the set {VI>V2,V3}' All energy levels up to the dissociation limit are included in the sum and conver-

gence is checked by repeating the calculation including levels only up to about 80% of the dissociation limit.

We also consider another, simpler approximate method23 for computing vibrational partition functions. In this approximation, only the ground vibrational state and the fundamental excitation energies (i.e., single-quantum excitations from the ground state) are used as input for the calculation. Defining A.m as the fundamental excitation energy of mode m and the ground state energy as EG

, the partition function is approximated by

.n!)PT exp ( - (3EG)

~vib (T)=n!=~[I-exp(-{3A.m)l· (48)

Then Wand A.m are calculated by second order perturbation theory, i.e., by Eqs. (42)-(46) with VI=v2=v3=0. This method is denoted SPT for "simple perturbation theory," because in this approximation, we use perturbation theory only for the ground state and single-quantum excitations-not for all of the energy levels. In contrast, full vibrational perturbation theory, using Eq. (47) but still assuming separation of rotations from vibrations and still stopping at second order, will be denoted PT ..

In order to obtain the force constants (and hence the spectroscopic constants) from the various potential energy surfag:s using SUR VIBTM, the potential must be defined on a grid for fitting purposes. In all cases, we calculate the potential energy on a 7 X 7 X 7 grid of points which are equally spaced along the internuclear distances (R I,R2) and angle (cp) (see Fig. 1) with the grid centered at the equilibrium geometry. The grid spacing is 0.04ao for RI and R2 and 5° for cp (where ao is the Bohr radius57 ). We

_have.found that varying the density of grid points (i.e., varying the number of points over a given range of internuclear distances) from a 5 X 5 X 5 grid to an 11 X 11 X 11 grid produced very little difference in the force constants (less than 1 cm - 1) for all of the molecules considered in this paper. Similar convergence tests were carried out to explore the dependence of the vibrational perturbation theory calculations on the range of internuclear distances used, and we find that the present calculations are well converged with respect to this range. We also calculated the force constants with the POL YRA TE program, 58 and slightly larger differences were found in the anharmonic force constants.

We note that Bartlett et al. 59 used the same program to study water with ab initio electronic structure calculations. They used a 5 X 5 X 5 grid of points with 0.03ao grid spacing for the bond length and 3° for the bond angle. When the same grid is used for the analytic potential functions in the present-work, we found errors less than 1 cm -1 in the frequencies and the cubic force constants of Eq. (41) and less than 5 cm -I in the quartic force constants. These errors, however, lead to an error in the anharmonic constants up to 28 cm -I. We conclude that it is essential to find a stable fit region to ensure quantitative accuracy. (Of course it is much easier to experiment with the grid parameters when using an analytic potential energy function, as here, than when using ab initio electronic structure calculations.)



Topper et al.: Quantum steam tables

TABLE 1. Calculated vibrational frequencies and perturbation theory parameters for H20 and D20.a

- f VI -f V2 -f V3

Harmonic ZPEK D.lh D.2h D.3h

Anharmonic ZPEi

Eoi xIIi xli XI3i

X22i

X23i

X33i

3834 1647 3935 4709

3656 1596 3745 4631

-0.16 -42.91 -16.83

-167.15 -17.72 -14.10 -49.58

H 20 (Ref. 39)C

3660 1594 3753

H2O (MKH)d

3834 1647 3935 4709

3573 1589 3658 4612

14.88 -63.24 -24.53

-244.21 -17.80 -20.98 -72.12

H2O Dp (Expt.)e (KH)

3834 2759 1648 1208 3935 2883

n.a. 3425

3657. 2667 1595 1180 3756 2779

n.a. 3383

n.a. -0.63 -42.58 -21.95 -15.93 -7.90

-165.82 -87.20 -16.81 -9.79 -20.32 -7.14 -47.57 -28.16

a( •.• ) indicates that the value was not reported; n.a. indicates that the value is not available.

DzO (Ref. 39)

2669 1179 2785

DP (MKH)

2759 1208 2883 3425

2623 1177 2731 3372

7.66 -32.21 -12.19

-126.91 -9.79

-10.90 -40.41

4997

DP (Expt.)

2784 1206 2889

n.a.

2668 1178 2788

n.a.

n.a. -22,58 -7.58

-87.15 -9.18

-10.61 -26.15

bCalculated from the Kauppi-Halonen (KH) potential, using a normal-mode analysis to compute vm and perturbation theory to compute D.m. cCalculated values from Ref. 39, obtained by those authors via variational methods. dThe same as b, except that the modified Kauppi-Halonen (MKH) surface is used. <Experimental values summarized in Refs. 41 and 48. fHarmonic frequency (in em-I). SIn dependent harmonic normal mode approximation to zero point energy (in em -I). hFundamental transition frequency (in em -I). iAnharmonic zero point energy computed via vibrational perturbation theory (in em-I). iparameters for Eq. (44), either computed using SURVIBTM (Ref. 49) or obtained from experiment (Refs. 39-41 and 48).

VI. RESULTS

VI. A. Perturbation theory transition energies and spectroscopic parameters

In Table I, we present harmonic frequencies and transition frequencies for H20 and D20 as computed from the potential energy surfaces summarized in the previous section. We also present the spectroscopic constants obtained from the perturbation theory calculations, as well as exper-

imental values for these constants.39-41,48 Considering first the H20 and D20 molecules, we find that the fundamentals we have obtained from the Kauppi-Halonen surface using vibrational perturbation theory agree well with the fundamentals computed by Kauppi and Halonen39 using variational methods for the vibrations. However, the fundamentals we have calculated from the modified KauppiHalonen surface40 for these molecules are somewhat

TABLE II. Calculated vibrational frequencies and perturbation theory parameters for HzS and H2Se. a

H2S (KH)

VI 2719 V2 1214 V3 2736 Harmonic 3335 ZPE D.I 2615 D.2 1183 D.J 2628 Anharmonic 3291 ZPE Eo 3.89 XII -23.59 xI2 -19.57 XI3 -95.01 X22 -5.43 X23 -21.20 x33 -25.29

H 2S (Ref. 39)

2615 1183 2628

H2S (Expt.)

2722 1215 2733

n.a.

2614 1183 2629

n.a.

n.a. -25.09 -19.69 -94.68 -5.72

-21.09 -24.00

H2Se (KH)

2437 1055 2452 2972

2343 1033 2358 2935

4.80 -21.49 -17.75 -83.32 -2.18

-17.85 -21.56

'See Table I for an explanation of rows and columns.


H 2Se (Ref. 39)

2344 1033 2357

H 2Se (Expt.)

2435 1054 2448

n.a.

2344 1034 2358

n.a.

n.a. -21.4 -17.7 -84.9 -2.4

-20.2 -21.7



TABLE III. Calculated vibrational partition functions for H2O.'

(!vl~(T) (fv~T( T) (1vi~( T) (fv~T(T) (1vi~(T) (1vi~(T) TCK) (KH,MKH)b (KH)C (KH)d (MKH)e (MKH)f (KH vs MKH)g

200 1.95X 10- 15 3.40 X 10- 15 3.40X 10- 15 3.89X 10- 15 3.89X 10- 15 14% 300 1.56 X 10- 10 2.26X 10- 10 2.26XlO- 1O 2.48 X 10- 10 2.48 X 10- 10 9% 400 4.42X 10-8 5.85X 10-8 5.8SX 10-8 6.26X 10-8 6.26X 10-8 7% 600 1.27X 10-5 1.S4x 10-5 1.S4X 10-5 1.61 X 10-5 1.61 X 10-5 5%

1000 1.27X 10-3 1.43 X 10-3 1.44x 10-3 1.48X 10-3 1.48X 10-3 3% 1500 1.4SX 10- 2 l.59X 10-2 1.60X 10-2 1.63X 10-2 1.64X 10- 2 6% 2400 1.16x 10-1 1.27X 10- 1 1.29X 10- 1 l.31X 10- 1 l.33X 10-1 10% 4000 7.26X 10-1 8.26X 10-1 8.12X 10- 1 7.99X 10-1 8.S0X 10:"1 12%

'All partition functions calculated with zero of energy at the equilibrium geometry. blndependent mode harmonic oscillator partition function from KH and MKH potential surfaces (see Tables I-II). "Simple perturbation theory (SPT) partition function from KH potential surface. dperturbation theory (PT) partition function from KH potential surface. "The same as c, except for the MKH surface. fThe same as d, except for the MKH surface. gThe difference between vibrational partition functions on the two surfaces calculated at the PT level. Average=6%.

different from those calculated from the Kauppi-Halonen surface; in particular, there are sizeable differences in the Lli and Ll3 fundamentals. Although the two potential surfaces have exactly the same harmonic frequencies, the spectroscopic constants for the two surfaces are very different.

Table II presents similar results for the H2S and H 2Se molecules. We see that the calculated fundamentals are in excellent agreement with the values calculated by Kauppi and Halonen,39 who used variational methods. The good agreement between our calculations and the ones carried out by Kauppi and Halonen is a good check.

VI. B. Separable vibrational and rotational partition functions

We present vibrational partition functions for all four molecules in Tables III-VI, using the harmonic oscillator model, perturbation theory, and the simple perturbation theory expressions given above, and in Table VII we give the classical rigid rotator partition functions for all four molecules. We see in Tables III and IV that, although the vibrational partition functions for the two water surfaces are identical in the harmonic oscillator approximation, there is a quantitative difference between the partition functions from the two surfaces at the perturbation theory

TABLE IV. Calculated vibrational partition functions for D20.'

(!vl~(T) (fv~T(T) (1vi~( T) TCK) (KH,MKH) (KH) (KH)

200 1.99 X 10- 11 2.70X 10- 11 2.70X 10- 11

300 7.36xlO-8 9.04X 10-8 9.04X 10-8

400 4.S2XlO-6 S.27XlO-6 S.28xlO-6

600 2.87Xl0-4 3.19xlO-4 3.20XlO-4

1000 9.09xl0- 3 9.81 X 10-3 9.83XIO-3

1500 6.26xlO-2 6.70XIO-2 6.74X 10-2

2400 3.74XI0-1 4.01XlO- 1 4.08 X 10-1

4000 2.04 X 10° 2.20XIO° 2.17X 10°

level. For H 20, the average unsigned difference is 6%, with the largest unsigned difference (14%) at T = 200 K. Since the perturbation theory and simple perturbation theory results are identical at this temperature, we infer that the discrepancy is due to the difference in the perturbation theory estimates of the zero point energy on the two surfaces. This is confirmed by a comparison of the harmonic partition function to the one obtained by perturbation theory for each potential surface; there is a difference of 20% between the harmonic and perturbation theory partition functions for H20 on the Kauppi-Halonen surface, and a difference of 26% for H 20 on the modified KauppiHalonen surface. The effect is similar for D 20 over the same temperature range, with an average unsigned difference of less than 4% between the two potentials at the perturbation theory level; here again, the largest unsigned difference between the two surfaces (8%) is at T = 200 K.

From Table V, we see that the harmonic oscillator vibrational partition function for H 2S at T=200 K differs by 27% from the perturbation theory partition function, which is the largest discrepancy between the two approximations we observe in this study. In Table VI, we see that the harmonic oscillator partition function also disagrees significantly with the perturbation theory result for H 2Se (by 23% at T=200 K).

(fv~T(T) (1vi~(T) ~(T) (MKH) (MKH) (KH vs MKH)b

2.89X 10- 11 2.92 X 10- 11 8% 9.46 X 10-8 9.51 X 10-8 5% S.46x 10-6 S.48xlO- 6 4% 3.28X 10-4 3.28xlO-4 2% 9.98XlO- 3 1.00 X 10-2 2% 6.82X 10-2 6.88X 10-2 2% 4.lOX 10-1 4.19xlO- 1 3% 2.26xlO° 2.26x 10° 4%

aSee Table III for an explanation of columns. One hundred and fifty eight bound energy levels are used in the perturbation theory calculations. ~he difference between vibrational partition functions on the two surfaces calculated at the PT level. Average=4%.

J. Chem. Phys., Vol. 98, No.6, 15 March 1993



TABLE V. Calculated vibrational partition functions for H2S.'

T(K) Q;1~(T) (fv~T(T) {tvi~(T) ~~(T) vs {tvi~(T)b

200 3.82X 10- 11 5.23XlO- 11 5.23X 10- 11 27% 300 1.14X 10-7 1.40 X 10-7 1.40 X 10-7 19% 400 6.26X 10-6 7.33XlO-6 7.33 X 10-6 17% 600 3.57XlO-4 3.99XlO-4 3.99XlO-4 11%

1000 l.04X 10-2 l.13x 10- 2 1.13 X 10-2 8% 1500 6.91 X 10-2 7.43 X 10- 2 7.47 X 10-2 7% 2400 4.04 X 10- 1 4.36X 10- 1 4.42 X 10- 1 9% 4000 2.18X 100 2.37XlO° 2.3 XlO° 5%

'See Table III for an explanation of unlabeled columns. bDeviation of harmonic oscillator partition functions from perturbation theory partition functions.

Overall, the agreement between the SPT and PT partition functions is within 1 % for all four molecules over the range 2oo<T<15oo K, with mOdestly larger differences at 2400 K and differences up to 6%-8% at 4000 K. This is a very important conclusion because simple perturbation theory may be applied without convergence checks with respect to the number of energy levels.

VI. C. Fourier path-integral Monte Carlo convergence checks

We next describe the parameters used in the Fourier path-integral Monte Carlo calculations. In the pathintegral calculations presented here, the potential energy is integrated along each sampled path using either 50- or loo-point Gauss-Legendre32 quadrature. We have carried out convergence studies for H20 on the modified KauppiHalonen potential energy surface (described below) at 1000 K by using 105 samples and varying the number N q of quadrature points from 50 to 75, 100, and 150 with K = 128 and 256 Fourier coefficients per degree offreedom. We find that the results are identical to well within the statistical error limits. We note that similar values of N q and K were found to be adequate in our previous study of the HCI molecule. 12 Convergence studies of H2S and H2Se at this temperature led to the same conclusion, i.e., 50 quadrature points and 128 Fourier coefficients per degree of freedom are adequate'to achieve the same level of convergence as that achieved for H20 at 1000 K, and we infer that these parameters are adequate to achieve good convergence at higher temperatures as well.

TABLE VI. Calculated vibrational partition functions for H2Se.'

T(K) (!v!~(T) (fv~TCT) {tvICT) ~~(T) vs {tvICT)

200 5.20X 10- 10 6.7SXlO- 1O 6.7SXlO- 1O 23% 300 6.51XlO-7 7.7SX10-7 7.7SXlO-7 16% 400 2.33XlO-5 2.66X 10-5 2.66XlO-5 12% 600 8.79XlO-4 9.64XlO-4 9.6SX 10-4 9%

1000 1.89X 10-2 2.03 X 10-2 2.03 X 10-2 7% 1500 1.11 X 10- 1 1.19 X 10- 1 1.20 X 10- 1 8% 2400 6.08XlO- 1 6.S3X 10- 1 6.57Xl0- 1 7% 4000 3.18X 100 3.44X 10° 3.2 XlO° 1%

"See Tables III and V for an explanation of columns.

We found that 128 Fourier coefficients per degree of freedom was sufficient to obtain good convergence (better than 5%) at T> 1000 K. However, similar studies of Hp

. at 600' K showed that more Fourier coefficients were needed to achieve good convergence at lower temperatures. In Table VIII, we present the results of a convergence study with respe~t to the number K of Fourier coefficients at a temperature of 600 K, using 106 samples and 50 quadrature points for each calculation. We find that adequate convergence may be achieved at this temperature by using K =256. This is also consistent with our previous studies. 12

Several additional parameters must be specified when using the adaptively optimized stratified sampling (AOSS) algorithm5

,12 to sample the configuration space. The AOSS algorithm proceeds in three stages: (1) optimization of the "great hypersphere" which constitutes the outermost boundary of the domain D; (2) optimization of the three stratification boundaries; and (3) optimal sampling of the strata defined in stage 2. In stage 1, the outer boundary radius was optimized by allowing a maximum of 12 trial hyperspheres with hyperradii equally spaced on the interval from 3 to 4ao for all calculations with T> 600 K; for T = 600 K, the stage 1 result at 1000 K was used as the hyperradius for the great hypersphere. In stage 2, 100 bins of equal volumes within the optimized hypersphere were sampled until 0.40n samples were accumulated, and these were used to optimize the stratification boundaries. A sam.ple size of 0.06n was used in each round of stage 3 until n samples were generated. By examining several test cases, we found no appreciable dependence of our results on small changes in these parameters; most notably, using larger values of the outermost hyperradius did not affect the calculations.

All error estimates are quoted at a level of two standard deviations. As the algorithm uses uncorrelated sampling in both the configuration and Fourier coefficient spaces, the error bars are easily estimated and are expected to be accurate estimates (to one or two significant figures) of 95% statistical confidence intervals. Thus, since the statistical errors dominate those due to finite N q and K, there is approximately a 95% chance that the true partition functions lie within the quoted error estimates. We use either 105 or 106 samples in each calculation as required to converge the calculation to the desired precision; in particular, we used enough sampling points so that the resulting statistical error limits are < 10% of the central value for all calculations.

VI. D. Comparison of vibrational-rotational partition functions

In Tables IX-XII, we present the results of the Fourier path-integral Monte Carlo calculations and compare them to the approximate methods discussed in Secs. V and VI B.

The top section of Table IX presents calculations for H20 on the Kauppi-Halonen surface. Examining first the results which assume an approximate separation of vibrations from classical rigid rotations, we see that the harmonic osCillator/rigid rotator approximation is in quanti-

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tative disagreement with both of the perturbation theory methods, which agree with one another quantitatively. Next examining the FPI calculations, we see that they generally agree with the perturbation theory calculation to within the statistical error limits for T<1500 K.

For temperatures at and above 2400 K, a noticeable discrepancy between the path-integral diculation and either of the perturbation theory results begins to appear (i.e., the difference between the two results is outside of the 95% confidence interval). We have carried out a convergence study of the Fourier path-integral calculations on this potential surface at 2400 K (see Table IX), and we are confident that our answer is well converged, and therefore the perturbation theory/classical rigid rotator model seems to be inaccurate in this temperature regime. This discrepancy may be due to the onset of appreciable vibrationalrotational coupling, or it may be due to the simple method we have used to model the rotations. However, the discrepancy does not seem to be due to the onset of appreciable dissociation, as the perturbation theory calculations at 2400 K are well converged (using 91 bound energy levels instead of using 119 levels yields a partition function at 2400 K which differs by less than 0.01 % from the larger calculation). A similar check at 4000 K shows a 0.3% difference between using 119 and 91 levels, so the perturbation theory results are well converged at this temperature as well.

Perturbation theory appears to be less accurate for the modified Kauppi-Halonen surface, for which results are presented in the bottom section of Table IX.

The calculations for D20 on the two surfaces are presented in Table X. For this molecule, the fractional statistical errors achieved by the Fourier path-integral method are generally smaller than for H 20. We see that the Fourier path-integral calculations are in excellent agreement with the perturbation theory model for T<1500 K on the Kauppi-Halonen surface, but for T>2400 K, we again see a noticeable discrepancy between the two results. However, the agreement between the Fourier path-integral calculations and the perturbation theory calculations on the modified Kauppi-Halonen surface is rather poor, with the perturbation theory calculations always well outside of the 95% statistical confidence limits over the entire temperature range. The perturbation calculation is converged with respect to the number of vibrational states included in the calculation to within 1 % at 4000 K and even better at lower temperatures. Thus the high-temperature difference of the accurate and approximate calculations may indicate the breakdown of the model we use for the rotational motions for this molecule or indicate that excited-state centrifugal distortions are large on this surface. Further study is necessary to fully evaluate the causes for the discrepancy.

In Table XI, we give results for H2S, computed using the Kauppi-Halonen potential energy surface for this molecule. The fractional statistical errors are somewhat smaller for this molecule than for H 20 and D 20, and the general trends are similar, i.e., the perturbation theory/ classical rigid rotator calculations agree closely with the

TABLE VII. Calculated classical rigid rotator partition functions Q;;'~R(T).

TCK) H2O D 20 H2S H2Se

200 2.351X 101 6.189X 101 6.834X101 9.170X101

300 4.320x Wi 1.137X 102 1.256 X 102 1.685 X 102

400 6.651X 101 1.751 X 102 1.933 X 102 2.594X102

600 1.222 X 102 3.216X 102 3.551X 102 4.765 X 102

1000 2.629X102 6.920X102 7.641 X 102 1.025 X 103

1500 4.830X102 1.271 X 103 1.404 X 103 1.883 X 103

2400 9.775 X 102 2.573X 103 2.841 X 103 3.812X 103

4000 2.103 X 103 5.536X 103 6.113 X 103 8.202 X 103

FPI calculations for T <2400 K. The convergence of the perturbation calculation with respect to the number of levels included is better than 2 % at 4000 K.

Table XII gives results for H 2Se, again computed using a Kauppi-Halonen potential energy surface. We see that the fractional statistical errors achieved by the FPI method are slightly less than those for H 2S. The discrepancy between the accurate FPI method and approximate perturbation theory calculations is noticeable (6%) at 1500 K for H2Se, and it is quite large (22%) at 4000 K. The perturbation calculation is well converged (about 2% at 4000 K, and within 0.2% for lower temperatures) in the case of H2Se.

VI. E. Quantum free energies

The quantum partition functions presented in the previous section may be used to compute free energies for an ideal gas. 1- 5,24,25 In the present section, we consider free energy functions in the form tabulated in the JANAFtables.28 In particular, we tabulate the Gibbs free energy function called gef( T) and defined as25

gef(T) == - (G-W)/T

=R In Qtrans(T)+R In Q(T),

==geftrans( T) + gefint ( T),

(j(T) =Q(T)ef3If1,

(49)

(50)

(51)

where gef( T) has been separated into translational and internal components. Here W is the energy of the ground

TABLE VIII. Convergence of vibrational-rotational partition functions at 600 K for H20 on the Kauppi-Halonen surface.'

J<b (Q(T»n±2wnC

64 (2.32±0.20) X 10-3

128 (2.26±0.20) X 10-3

256 (1.91 ±0.16) X 10-3

512 (2.01 ±0.17) X 10-3

"Fifty Gauss-Legendre quadrature points used in all calculations with 106

samples per calculation . . i>yhe number of Fourier coefficients per degree of freedom used in calcu

lations. cAOSS-U Fourier path-integral Monte Carlo calculation of the coupled partition function. The error bars for the calculations are estimated at the 95% confidence level.



Topper et at.: Quantum steam tables 5001

TABLE IX. Calculated vibrational-rotational partition functions for H20 on the Kauppi-Halonen surface."

T(K) Q!1~~~R (fv~T ifro~R (fvi~{to~R (Q(T»n±2wnb FSEc nd K" N f

a

Kauppi-Halonen surface 600 1.56x 10-3 1.88X 10-3 1.88X 10-3 (2.0±0.2) X 10-3 10% 106 512 50

1000 3.34X 10- 1 3.77X 10- 1 3.77X 10- 1 (3.9±0.4) X 10- 1 10% 105 128 100 1500 6.98XIOO 7.69XIOO 7.72X 10° (7.6±0.5) X 10° 7% 105 128 100 2400 1.14 X 102 1.24X 102 1.26 X 102 (1.34±0.6) X 102 5% 105 128 100 2400 (1.32±0.02) X 102 2% 106 128 50 2400 (1.34±0.02) X 102 2% 106 256 50 2400 (1.33±0.02) X 102 2% 106 256 100 4000 1.53 X 103 1.68X 103 1.71 X 103 (1.89±0.06) X 103 3% 105 128 100

Modified Kauppi-Halonen surface 600 1.56X 10-3 1.97X 10-3 1.97 X 10- 3 (2.3±0.2) X 10-3 9% 106 256 50

1000 3.34X 10- 1 3.88X 10- 1 3.89X 10- 1 (4.0±0.3) X 10- 1 8% 105 128 100 1500 6.98XlOo 7.88XlOo 7.92 X 10° (8.6±0.5) X 10° 6% 105 128 100 2400 1.14X 102 1.28 X 102 1.30 X 102 (1.38±0.06) X 102 4% 105 128 100 4000 1.53X 103 1.74X 103 1.79 X 103 (2.1O±0.06) X 103 3% 105 128 100

'All partition functions calculated with zero of energy at the equilibrium geometry. The separation of rotations from vibrations is assumed and the rotations are treated as classical and rigid. See Table III for explanations of columns 1-4.

bAOSS-U Fourier path-integral Monte Carlo calculation of coupled partition function. The error bars for the calculations are estimated at the 95% confidence level.

<Fractional statistical errors (FSE) at the 95% confidence level. dThe number of samples used in Fourier path-integral Monte Carlo calculations. "The number of Fourier coefficients used in Fourier path-integral Monte Carlo calculations. fThe number of Gauss-Legendre quadrature points used in Fourier path-integral Monte Carlo calculations.

vibrational-rotational state and Q( T) is the internal molecular partition function with the zero of energy shifted to this level. We will discuss the free energy function of Eq. ( 49) because of its prominent role in the literature and in summarizing experimental data. We note, however, that to estimate thermodynamic data (such as equilibrium constants) from potential energy or electronic structure data requires both Wand s..ef(T), i.e., it requires information about Q( T), not just Q( T) in which W cancels out.

Equation (49) was used to estimate gef( T) in the harmonic approximation, by simple and full vibrational perturbation theory, and from Fourier path-integral Monte £arlo calculations. W is calculated harmonically to obtain Q( T) from Q( T) in the harmonic approximation, and it is calculated from perturbation theory in the other cases. The resulting gef( T) values are in Table XIII for H20 and in Table XIV for D20, H2S, and H2Se. In the case of H20, a

recent calculation by Martin, Fran~ois, and Gijbels25 is also included in the comparison. Additional values of gef( T) from sets of standard thermochemical tables27,60,61 are also included.

Due to its importance, the free energy function of H20 has been studied extensively. The most accurate previously available values are those of Martin and co-workers,21 who used ab initio electronic structure methods62,63 to obtain a quartic force field for H20; they treated rotation by an approximate nonseparable quantum mechanical method24

including centrifugal distortion, and they obtained the vibrational partition functions by perturbation theory.46-54 [They did not include Eo in their calculations, but it has no effect on gef( T) because EG is substracted from G in Eq. (49).] We find that the ab initio results of Martin and co-workers25 are practically identical to those we have obtained from the Kauppi-Halonen surface via well con-

TABLE X. Calculated vibrational-rotational partition functions for D20 on the Kauppi-Halonen surface."

T(K) ~~~~R (fv~TQfo~R {fvi~Q;,~R (Q(T)n±2wn FSE n K Na

Kauppi-Halonen surface 600 9.24XIO-2 1.03 X 10- 1 1.03 X 10- 1 (1.08±0.05) X 10- 1 5% 106 256 50

1000 6.29XlOo 6.79X 10° 6.80XlOo (6.8±0.5) X 10° 7% 105 128 100 1500 7.96X 101 8.52X 101 8.57XIO I (8.3±0.5) X 101 6% 105 128 100 2400 9.63X102 1.03 X 103 1.05 X 103 ( 1.08 ± 0.04 ) X 103 4% 105 128 100 4000 1.13 X 104 1.22 X 104 1.20 X 104 (1.38 ±0.03) X 104 2% 105 128 100

Modified Kauppi-Halonen surface 600 9.24XIO-2 1.05 X 10- 1 1.06 X 10- 1 (1.19±0.06) X 10- 1 5% 106 256 50

1000 6.29 X 10° 6.91 X 10° 6.93 X 10° (7.7±0.5) X 10° 7% 105 128 100 1500 7.96X 101 8.67X 101 8.75X 101 (9.5±0.5) X 101 5% 105 128 100 2400 9.63 X 102 1.05 X 103 1.08 X 103 (1.17±0.04) XJ03 3% 105 128 100 4000 1.13X 104 1.25 X 104 1.25 X 104 (1.49±0.03) X 104 2% 105 128 100

'See Table IX for an explanation of rows and columns.



TABLE XI. Calculated vibrational-rotational partition functions for H2S."

T(K) Q!f~~~R {fv~T~:R {ti~Q;;,~R

600 1.27 X 10- 1 1.42 X 10- 1 1.42 X 10- 1

1000 7.9SXlO° 8.60X 10° 8.61 X 10° ISoo 9.70X 101 1.04 X 102 1.04 X 102

2400 1.15 X 103 1.24 X 103 1.26 X 103

4000 1.33 X 104 1.4SX 104 1.4 X 104

"See Table IX for explanations of rows and columns.

verged Fourier path-integral calculations for T.;;;2400 K. This is very encouraging for their ab initio approach.

Empirical estimations of gef( T) are taken from the JANAF tables28 and from Gurvich et al. 60 For the tabulation of Gurvich et aI., the vibrational partition function is obtained by a summation of the form

m

where Em is evaluated from an expression like Eq. (42), but including more terms, e.g., the Ymna(vm+!) (Vn+!) (va +!) terms, and where the constants were obtained from experimental vibrational and rotational spectra. Note that Eq. (52) does not assume separable vibrational and rotation. The rotational partition functions Qrot are calculated using a rigid asymmetric top approximation with the Stripp-Kirkwood correction and centrifugal distortion corrections. The JANAF28 tables are based on a similar, but somewhat more complicated procedure. Both treatments28

,60 take account of Darling-Dennison resonance. At all the temperatures, the results from both tables agree very well with the Fourier path-integral calculations. However, the harmonic and perturbation theory results deviate from the other results in Table XIII at 2400 K and higher. The error in the perturbation results may be due to the exclusion of the effect of centrifugal distortion on the rotational partition function since similar formalisms for the vibrational partition functions are used in the perturbation theory and empirical calculations, and the parameters (i.e., the frequencies, fundamentals, etc.) are almost the same in the perturbation theory and empirical calculations.

In the case of D 20, similar procedures as for H 20 were used by Gurvich et al. 60 and in JANAF28 tables. The conclusions we derive from the comparisons are also generally similar to those for H 20. At 1500 and 2400 K, the results from Gurvich et al. are not within the 95% confidence range of the path-integral calculations, but if we estimate the errors in their tabulation by interpolating their stated

TABLE XII. Calculated vibrational-rotational partition functions for H 2Se."

T(K) ~~(£~R ~r.;~~R {tJ;Q~R

600 4.19X 10- 1 4.60XlO- 1 4.60xlO- 1

1000 1.94 X 101 2.08 X 101 2.08X101

1500 2.09 X 102 2.24X102 2.25x 102

2400 2.32XW 2.49 X 103 2.5 X 103

4000 2.61 X 104 2.82X104 2.6 X 104

"See Table IX for explanations of rows and columns.

(Q(T)n±2wn FSE n K Na

(1.46±0.06) X 10- 1 4% 106 256 SO (9.4±0.7) X 10° 7% lOs 128 100 (1.11 ±0.06) X 102 5% lOs 128 100 (1.27 ± 0.04) X 103 3% lOs 128 100 ( 1.67 ± 0.04 ) X 104 2% lOs 128 100

errors at lower and higher temperatures, then the results do agree with the sum of the error bars. A significant discrepancy between the path-integral results and the results of the harmonic and the perturbation calculations occurs only above 2400 K.

There is less data in the literature28,60,61 for the Gibbs free energy functions for H2S and H 2Se than for the previoustwQ cases. This is especially true in the case of H2Se, where no reliable estimation is available at high temperatures. Even where tabulated data exist at lower temperatures for this molecule, they were calculated from the entropy and the heat capacity (Cp ) which were derived from molecular constants61 and are less accurate than the present calculations since they are based on limited data. The present results from both perturbation theories are quite close to the path-integral calculations, while the results from the harmonic approach are not within the 95% confidence range of the path-integral calculations at 1500 K and higher. It is reasonable to assume that the present quantal results for H2Se are the most accurate results available for this molecule.

VII. CONCLUSIONS

By producing accurate quantum mechanical partition functions, including vibration-rotation coupling, modemode coupling, and anharmonicity, we have been able to test the perturbation theory/classical rigid rotator model for triatomics more systematically than before. We have found that for H20, D20, H2S, and H 2Se, this model is very accurate for the calculation of vibrational-rotational partition functions for temperatures in the range 600.;;;T.;;;1500 K.

The fact that the perturbation theory/rigid rotator model gives such accurate values of the partition function for H20, D20, H2S, and H2Se for the 600,;;;T,;;;1500 K temperature range leads to the conclusion that a classical approximation to the rotations does not produce apprecia-

(Q(T)n±2wn FSE n K Na

(4.7±O.2) X 10- 1 4% 106 256 50 (2.1 ±0.1) X 101 5% lOS 128 100 (2.4±0.1) X 102 4% lOS 128 100 (2.67±0.09) X 103 3% lOS 128 100 (3.33±0.08) X 104 2% 105 128 100

J. Chem. Phys., Vol. 98, No.6, 15 March 1993



TABLE XIII. Quantum Gibbs free energy functions for H2O.'

ge~nt(T) gef(T) gefint(T) T geftrans( T)b FPIc FPlc,d MFGe

600 138.7 40.7 179.4 :J:8:~ 179.1 1000 149.3 47.6 196.9:J:8:~ 196.9 1500 157.7 53.8 211.5:!:g:~ 212.0 2400 167.5 63.7 231.2:J:8:i 231.1 4000 178.1 76.6 254.7:J:8:~

gef(T) gef(T) JANAF (Ref. 60)

179.0 179.0 196.8 196.8 211.9 212.0 231.1 231.2 254.5 254.9

gef(T) SPTf

178.8 196.6 211.6 230.7 253.7

gef(T) PTg

178.8 196.6 211.6 230.8 253.8

gef(T) HOh

178.7 196.5 211.4 230.3 253.2

'See Eq. (51) for a definition of free energy functions. Units of gef( T) are J (mol K) -1.

'1'ranslational contribution at P= 1 bar= 100 000 Pa. CObtained via AOSS-U Fourier path-integral Monte Carlo from the Kauppi-Halone surface. dThe range of 95% confidence estimated from 2wn in Table I.X. <From Ref. 25. rObtained via simple vibrational perturbation theory from the Kauppi-Halonen surface. 'Obtained via vibrational perturbation theory from the Kauppi-Halonen surface. hObtained via the harmonic oscillator model from the Kauppi-Ha1onen surface.

ble errors in the vibrational-rotational partition function for these molecules in this particular temperature regime. It should be noted that one could alternatively use more sophisticated methods to estimate the rotational partition function,24 but the accuracy achieved by using the classical rigid rotator model seems adequate for thermochemical purposes at least for the molecules studied here.

The improvement in the treatment of vibrations by including anharmonicity through second order is very significant, while the harmonic oscillator/rigid rotator model disagrees strongly with the Fourier path-integral calculations (by an average unsigned error of 16% overall, with individual differences as large as 32% ), the vibrational perturbation theory/rigid rotator model is in good overall agreement with the Fourier path-integral calculations (with the average unsigned discrepancy being 7% overall and with the largest individual difference being 22%). Further comparisons of this kind for a wider variety of systems are necessary to address whether this conclusion may be valid for other cases.

The Fourier path-integral Monte Carlo method used to obtain the fully coupled quantal results is more slowly con-

vergent at lower temperatures, and so we used larger sample sizes at 600 K. However, the statistical properties of the Monte Carlo method improve rapidly as the temperature is increased, i.e., there is a decrease in the fractional statistical errors for a given sample size. As a result, in the same temperature range where the perturbation theory/rigid rotator model appears to begin to break down, the Fourier path-integral method performs extremely well, achieving statistical errors on the order of 2%-3% or less, while using as few as 105 samples. This behavior made it possible to carry out extremely accurate calculations of the fully coupled vibrational-rotational partition function in the high-temperature (T;;;'2400 K) regime. It should be recognized that the present calculations, although they already yield accuracy comparable to or better than standard thermochemical tables for interesting molecules, are still just prototype applications of a new method. The Fourier path-integral method can also be applied to floppy molecules, but the perturbation theory treatment is likely to break down for such systems. We propose that it would be very interesting to carry out studies of systems of this type in the future. Furthermore, with improved sampling

TABLE XIV. Quantum Gibbs free energy functions for D20, H2S, and H2Se.a

ge~nt(T) gef(T) gef(T) gef(T) gef(T) gef(T) gef(T) gef(T) T geftrans ( T) FPI FPI (JANAF) (Ref. 60) (Ref. 61) (PT) (SPT) (HO)

D20 600 140.0 48.9 188.9:!:8:! 188.7 188.7 188.5 188.5 188.5 1000 150.6 56.4 207.0:!:8:~ 207.2 207.2 207.0 207.0 206.9 1500 159.03 63.71 222.7:!:8:~ 223.2 223.3 223.0 223.0 222.7 2400 168.80 74.93 243.7:!:8j 243.8 243.9 ·243.4 243.4 243.0 4000 179.42 89.37 268.8:!:8:~ 268.6 268.7 267.7 267.7 267.3

H 2S 600 146.6 49.6 196.2:!:8:~ 196.1 196.2 196.1 196.0 195.9 195.9 1000 157.2 58.0 215.2:!:8:~ 214.6 214.7 214.8 214.5 214.5 214.3 1500 165.6 65.40 231.0:!:8:! 230.7 230.9 230.9 230.5 230.5 230.3 2400 175.4 75.8 251.2:!:8:~ 251.4 251.6 251.2 251.0 250.6 4000 186.0 90.7 276.7:!:g:~ 276.3 276.7 275.2 275.5 274.9

HzSe 600 157.6 52.2 209.8:!:8:l 209.3 209.6 209.6 209.6 1000 168.2 60.4 228.6:!:8:~ 228.3 228.5 228.5 228.4 1500 176.6 67.8 245.6:!:8:~ 244.6 245.0 245.0 244.7 2400 186.4 80.2 266.6:!:g:~ 266.1 266.0 265.6 4000 197.0 95.4 292.4:!:8:~ 290.3 291.0 290.4

aSee Table XIII for a description of columns. All calculations obtained using the appropriate Kauppi-Halonen energy surface.



procedures-which are definitely possible and which will be pursued in later work, we should be able to treat lower temperatures and floppy molecules.

It would be interesting to compare the present directevaluation technique to indirect techniques based on parameter integration.64 It is especially important to determine which method or combination of methods is most efficient for more anharmonic many-dimensional systems.

VIII. SUMMARY

We have carried out a series of calculations of quantum mechanical vibrational-rotational partition functions for four triatomic molecules using Fourier path-integral methods, fully including the effects of anharmonicity, vibrational mode-mode coupling, and vibrational-rotational coupling on the quantum density matrix. We compared the resulting partition functions over a range of temperatures (600-4000 K) to approximate forms which assume the separation of vibrational motions from rotational motions, treating the rotations as a classical rigid rotator and the vibrations in three different approximations: (i) the harmonic oscillator model; (ii) full second order perturbation theory; and (iii) a recently presented "simple perturbation theory" model, which uses the perturbation theory calculations for the zero-point energy and the fundamentals in a functional form similar to that used for a set of independent normal modes. We find that for all the cases considered here, the harmonic model is in quantitative disagreement (as much as 30%) with the Fourier path-integral calculations and with the perturbation theory calculations. The two perturbation theory models are in excellent agreement with one another throughout the 600-2400 K temperature range with differences up to 6%-8% at 4000 K, and they agree with the Fourier path-integral results within the statistical error limits for almost all of the cases considered up until the highest temperatures (2400 and 4000 K), at which small differences start to manifest themselves. Thus the accuracy of the perturbation theory / classical rigid rotator treatment appears to be excellent overall (with an average unsigned deviation of 7% from the Fourier path-integral result) for the systems considered. Thus success of the simple perturbation theory calculation is especially encouraging since this method requires only a subset of the quartic force field and very little computation beyond that.

For H 20, we studied the change in partition functions when the potential modified at the cubic level in previous work to improve the predictions for clusters is used instead of the accurate isolated-molecule potential of Kauppi and Halonen. The partition functions for H 20 and D 20 change by about 10% at high temperature.

We also used the accurate and approximate partition functions to compute free energy functions referenced to the ground state, and we compared these to standard thermochemical tabulations, when available. The present methods are found to yield accuracy comparable to or better than the available empirical data, so that quantum mechanical calculations must now be added to a statistical thermodynamics toolbox as a seriously competitive tech-

nique. By this we mean that they are affordable but more accurate than simpler theoretical models used in the past, and they are competitive in accuracy with experimental methods.

Because improvements in sampling efficiency are envisioned, we anticipate future applications to larger molecules as well.

ACKNOWLEDGMENTS

We wish to thank Dr. Gregory Tawa and Dr. Xin Gui Zhao for a number of helpful conversations. We also thank Mr. Destin Jume11e-y-Picokens for help with the literature search. This work was supported in part by the National Science Foundation. Mr. Jumelle-y-Picokens was supported by the National Institutes of Health through its Summer High School Minority Research Apprenticeship program.

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